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Smart move on replacing the blue mainspring with a white alloy. Much, much better spring. If you’ve located a good source for these please let the rest of us know about it! They’re getting expensive. If some of what I’m about to write seems obvious or redundant to you please understand that it’s for the benefit of others that may be starting out and want to follow this thread.
I would certainly get the winding problem straightened out before anything else. My feeling though is that this wouldn’t be your problem or at least the only problem with the watch running 3 mins. slow a day. The balance in your Elgin should remain fairly consistent whether it’s swinging larger arcs when fully wound or shorter arcs as the mainspring power drops off. Although if you are getting some slipping action of the mainspring and the spring you replaced is a bit too strong then periodic overbanking (balance rotating too far resulting in the roller jewel hitting the back of the pallet horn) is possible. That can really cause timing problems. So yes I would definitely fix the mainspring-barrel problem.
The fact that the rate is consistent in different positions helps narrow things down a lot. It pretty much eliminates the possibility that there is a problem with the hole jewels that you replaced or balance pivot problems. If the balance displays good motion from about 270 degrees when fully wound to around 230 degrees when power is low then I would check out the following things.
First: Too much space between the regulator pins. This is very often the problem. There should be very little clearance between the hairspring and pins when the balance is ocillating. The further apart the pins the longer the effective length of the hairspring…which means the watch will run slower. This can also affect the isochronism.
Second: Loose index pin or pins.
Third: Check under high magnification for a rust spot or spots on the hairspring. Also look at the hairspring up around stud to see if it has a blemish (on a blue spring). This could indicate a weak point where the spring has been bent too often. A weaker spring means a slower balance. The rust is a more common problem though.
Forth: Balance wheel out of true. Check that the outer end of the balance arms line up with the other end at the slit. If they are bent out at all then the diameter of the balance has been increased which will cause the watch to run slower.
Fifth: Loose roller jewel.
Sixth: Timing screws…we will get to this one if the others are ok or if we need to make an adjustment.
Anxiously awaiting your response!